Ground states of the planar nonlinear Schr\"odinger--Newton system with a point interaction

Abstract

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schr\"odinger--Newton system with a point interaction: \[ \begin{cases} - \Delta_\alpha u = w u + \beta u |u|^{p - 2} &\text{on} ~ \mathbb{R}^2; \\ - \Delta w = 2 \pi |u|^2 &\text{on} ~ \mathbb{R}^2; \\ \|u\|_{L^2}^2 = c, \end{cases} \] where $p > 2$; $\alpha, \beta \in \mathbb{R}$ and $- \Delta_\alpha$ denotes the Laplacian of point interaction with scattering length $(- 2 \pi \alpha)^{- 1}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem \[ \mathrm{i} \psi' (t) = - \Delta_\alpha \psi (t) - (\log |\cdot| \ast |\psi (t)|^2) \psi (t) - \beta \psi (t) |\psi (t)|^{p - 2}. \]